I did insert your sample records and ran the code, See attached. It works well.

If object_id('tempdb..#t') IS NOT NULL DROP TABLE #t;
CREATE TABLE #t( BeneficiaryID VARCHAR(10), EligYear INT, EligMonth INT );

INSERT INTO #t(BeneficiaryID, EligYear, EligMonth )
Select '0068576102',2016,3 UNION
Select '0068576102',2016,6 UNION
Select '0068576102',2016,7 UNION
Select '0068576102',2016,8 UNION
Select '0068576102',2016,9 UNION
Select '0068576102',2016,10 UNION
Select '0068576102',2016,11 UNION
Select '0068576102',2016,12 UNION
Select '0068576102',2017,10 UNION
Select '0068576102',2017,11 UNION
Select '0068576102',2017,12 UNION
Select '0078576103',2016,8 UNION
Select '0078576103',2016,9 UNION
Select '78576103',2016,8 UNION
Select '78576103',2016,9 UNION
Select '78576103',2016,10 UNION
Select '78576109',2016,8 UNION
Select '78576109',2016,9 UNION
Select '78576109',2016,11 UNION
Select '78576109',2016,12;

INSERT INTO #t(BeneficiaryID, EligYear, EligMonth )
Select '12345678',2016,6 UNION
Select '12345678',2016,7 UNION
Select '12345678',2016,8 UNION
Select '12345678',2016,9 UNION
Select '12345678',2016,10 UNION
Select '12345678',2017,3 UNION
Select '12345678',2017,4 UNION
Select '12345678',2017,5

;
WITH T AS
(
SELECT *
     ,DENSE_RANK() OVER (PARTITION BY BeneficiaryID, EligYear ORDER BY EligMonth) - EligMonth AS Grp2
FROM #t
)
--Select * FROM T ORDER BY 1
,
CONSECUTIVE_MONTHS_TOGETHER as
(
SELECT
     BeneficiaryID,
     EligYear,
     MIN(EligMonth) AS RangeStart,
   MAX(EligMonth) AS RangeEnd,
     MAX(EligMonth) - MIN(EligMonth) as DIFF
FROM T
GROUP BY BeneficiaryID,EligYear, Grp2
)
Select * FROM CONSECUTIVE_MONTHS_TOGETHER WHERE DIFF >= 2 --( See Explanation )

/*
This solution works when the numbers are in a sequence ( that is the case with my example )
So if you had any 3 months of consecutive coverage the difference between the largest and smallest must be > 1 -- OR ( >= (2) )
*/